How do I change the shape of an object while keeping all of its data?
You can use two different methods.

Switch the meshes

Select your cube. In the Data panel, the first field allows you to select another mesh :


The object data will be preserved (logic bricks, modifiers, etc). Only the mesh data will be affected (vertex groups, materials slots, etc)

Use parenting

This method is preferable if your character already has an armature, and allows to keep the cube Physics settings.

Parent the character to the cube : select the character (or its armature if it has one), keep Shift pressed, then select the cube, hit Ctrl+P, select Object.

Make the cube invisible : in its Physics panel, tick Invisible.

To "hide" the cube in the 3d view but keep it selectable and usable, you can set Maximum draw type : Wire in its Object panel.